A Power Series Solution About $x = 0$ Of the Differential Equation

question 16

Multiple Choice

A power series solution about $x = 0$ of the differential equation $y ^ { \prime \prime } - y = 0$ is

• A) \(y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ! + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 ) !\)
• B) \(y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 )\)
• C) \(y = c _ { 0 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ^ { 2 } + c _ { 1 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k + 1 } / ( 2 k + 1 ) ^ { 2 }\)
• D) \(y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) ! + c _ { 1 } \sum _ { k - 0 } ^ { \infty } x ^ { 2 k - 1 } / ( 2 k - 1 ) !\)
• E) \(y = c _ { 0 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k } / ( 2 k ) + c _ { 1 } \sum _ { k = 0 } ^ { \infty } x ^ { 2 k - 1 } / ( 2 k - 1 )\)

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Definitions:

Somatic Delusion

A false belief about one's body, including its functions or appearance, that is unfounded and persistent.

Referential Delusion

A false belief that events, objects, or other people in one's environment have a particular and unusual significance to oneself.

Weather Control

The act of manipulating or altering certain aspects of the environment to change the weather, often considered speculative or theoretical.

Auditory Hallucinations

The experience of hearing sounds like voices that do not actually exist. A common symptom of schizophrenia.